3.6.89 \(\int \frac {\cosh ^3(x)-\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx\) [589]

Optimal. Leaf size=33 \[ -\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3 (1+\tanh (x))} \]

[Out]

-4/9*arctan(1/3*(1-2*tanh(x))*3^(1/2))*3^(1/2)-1/3/(1+tanh(x))

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Rubi [A]
time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2099, 632, 210} \begin {gather*} -\frac {4 \text {ArcTan}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3 (\tanh (x)+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^3 - Sinh[x]^3)/(Cosh[x]^3 + Sinh[x]^3),x]

[Out]

(-4*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/(3*Sqrt[3]) - 1/(3*(1 + Tanh[x]))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {align*} \int \frac {\cosh ^3(x)-\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx &=\text {Subst}\left (\int \frac {1+x+x^2}{1+x+x^3+x^4} \, dx,x,\tanh (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{3 (1+x)^2}+\frac {2}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{3 (1+\tanh (x))}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{3 (1+\tanh (x))}-\frac {4}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tanh (x)\right )\\ &=-\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3 (1+\tanh (x))}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 37, normalized size = 1.12 \begin {gather*} \frac {1}{18} \left (8 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 \tanh (x)}{\sqrt {3}}\right )-3 \cosh (2 x)+3 \sinh (2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]^3 - Sinh[x]^3)/(Cosh[x]^3 + Sinh[x]^3),x]

[Out]

(8*Sqrt[3]*ArcTan[(-1 + 2*Tanh[x])/Sqrt[3]] - 3*Cosh[2*x] + 3*Sinh[2*x])/18

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Maple [C] Result contains complex when optimal does not.
time = 0.13, size = 78, normalized size = 2.36

method	result	size
risch	\(-\frac {{\mathrm e}^{-2 x}}{6}+\frac {2 i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {3}\right )}{9}-\frac {2 i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {3}\right )}{9}\)	\(44\)
default	\(-\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-1-i \sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}-\frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-1+i \sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x,method=_RETURNVERBOSE)

[Out]

-2/3/(tanh(1/2*x)+1)^2+2/3/(tanh(1/2*x)+1)+2/9*I*3^(1/2)*ln(tanh(1/2*x)^2+(-1-I*3^(1/2))*tanh(1/2*x)+1)-2/9*I*
3^(1/2)*ln(tanh(1/2*x)^2+(-1+I*3^(1/2))*tanh(1/2*x)+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (26) = 52\).
time = 1.39, size = 70, normalized size = 2.12 \begin {gather*} \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {1}{6} \, e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x, algorithm="maxima")

[Out]

4/9*sqrt(3)*arctan(1/6*3^(3/4)*sqrt(2)*(2*sqrt(3)*e^(-x) + 3^(1/4)*sqrt(2))) - 4/9*sqrt(3)*arctan(1/6*3^(3/4)*
sqrt(2)*(2*sqrt(3)*e^(-x) - 3^(1/4)*sqrt(2))) - 1/6*e^(-2*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
time = 0.77, size = 74, normalized size = 2.24 \begin {gather*} -\frac {8 \, {\left (\sqrt {3} \cosh \left (x\right )^{2} + 2 \, \sqrt {3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {3} \sinh \left (x\right )^{2}\right )} \arctan \left (-\frac {\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right )}{3 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) + 3}{18 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x, algorithm="fricas")

[Out]

-1/18*(8*(sqrt(3)*cosh(x)^2 + 2*sqrt(3)*cosh(x)*sinh(x) + sqrt(3)*sinh(x)^2)*arctan(-1/3*(sqrt(3)*cosh(x) + sq
rt(3)*sinh(x))/(cosh(x) - sinh(x))) + 3)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (34) = 68\).
time = 0.52, size = 102, normalized size = 3.09 \begin {gather*} \frac {4 \sqrt {3} \sinh {\left (x \right )} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sinh {\left (x \right )}}{3 \cosh {\left (x \right )}} - \frac {\sqrt {3}}{3} \right )}}{9 \sinh {\left (x \right )} + 9 \cosh {\left (x \right )}} + \frac {3 \sinh {\left (x \right )}}{9 \sinh {\left (x \right )} + 9 \cosh {\left (x \right )}} + \frac {4 \sqrt {3} \cosh {\left (x \right )} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sinh {\left (x \right )}}{3 \cosh {\left (x \right )}} - \frac {\sqrt {3}}{3} \right )}}{9 \sinh {\left (x \right )} + 9 \cosh {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)**3-sinh(x)**3)/(cosh(x)**3+sinh(x)**3),x)

[Out]

4*sqrt(3)*sinh(x)*atan(2*sqrt(3)*sinh(x)/(3*cosh(x)) - sqrt(3)/3)/(9*sinh(x) + 9*cosh(x)) + 3*sinh(x)/(9*sinh(
x) + 9*cosh(x)) + 4*sqrt(3)*cosh(x)*atan(2*sqrt(3)*sinh(x)/(3*cosh(x)) - sqrt(3)/3)/(9*sinh(x) + 9*cosh(x))

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Giac [A]
time = 0.99, size = 22, normalized size = 0.67 \begin {gather*} \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} e^{\left (2 \, x\right )}\right ) - \frac {1}{6} \, e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x, algorithm="giac")

[Out]

4/9*sqrt(3)*arctan(1/3*sqrt(3)*e^(2*x)) - 1/6*e^(-2*x)

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Mupad [B]
time = 0.37, size = 22, normalized size = 0.67 \begin {gather*} \frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^{2\,x}}{3}\right )}{9}-\frac {{\mathrm {e}}^{-2\,x}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)^3 - sinh(x)^3)/(cosh(x)^3 + sinh(x)^3),x)

[Out]

(4*3^(1/2)*atan((3^(1/2)*exp(2*x))/3))/9 - exp(-2*x)/6

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